Chapter 33, Problem 33.80CP

P33.80a shows a parallel RLC circuit. The instantaneous voltages (and rms voltages) across each of the three circuit elements are the same, and each is in phase with the current in the resistor. The currents in C and L lead or lag the current in the resistor as shown in the current phasor diagram, Figure P33.80b.

(a) Show that the rms current delivered by the source is

$I_{rms}=\Delta V_{rms}{\left[\frac{1}{R^2}+{\left(\omega C-\frac{1}{\omega L}\right)}^2\right]}^{1\backslash 2}$

(b) Show that the phase angle ϕ between ΔV_rms and I_rms is given by

$\tan \varphi =R\left(\frac{1}{X_c}-\frac{1}{X_L}\right)$